# Tree Diagrams - eccgcmaths.weebly.com The AND Rule The Probability of events A AND B happening: P(A and B) = P(A) x P(B) I flip two coins, what is the probability that I get two heads? P(HH) = 1 4 Coin 1 Coin 2 1 2 1 2 1 2 Probabilities on each branch add up to 1 Heads P(HH) = 1 x 1 = 1 2 2 4 Heads P(HT) = 1 x 1 = 1 1 2

Tails 1 2 Heads 2 2 4 P(TH) = 1 x 1 = 1 2 2 4 Tails 1 2 Possible outcomes are shown on the end of the branches Tails P(TT) = 1 x 1 = 1 2 2 Multiply along the branches to work out the probabilities

4 Helen passes through 2 sets of traffic lights on her way to work. The probability that she stops at the first set is 0.3 The probability that she stops at the second set is 0.2 What is the probability that she has to stop exactly once? = 0.24 + 0.14 Lights 1 Lights 2 0.2 Stops P(SS) = 0.3 x 0.2 = 0.06 0.8 Doesnt Stop P(SD) = 0.3 x 0.8 = 0.24 0.2 Stops P(DS) = 0.7 x 0.2 = 0.14 Doesnt Stop P(DD) = 0.7 x 0.8 = 0.56 Stops 0.3

0.7 Doesnt Stop = 0.38 0.8 I have a bag of sweets, 3 blue and 7 red. I pick one sweet out without looking, eat it and then pick out and eat a second sweet. What is the probability that I eat one blue and one red sweet? 42 7 = 90 15 Sweet 1 Sweet 2 6 9 7 10 3 10 Red P(RR) = 7 x 6 = 42 10 9

90 Red 3 9 Blue 7 9 Red P(RB) = 7 x 3 = 21 10 9 90 P(BR) = 3 x 7 = 21 10 9 90 Blue 2 9 Blue P(BB) = 3 x 2 = 6 10

9 90 Tuesday Monday 0.3 It rains Vertically, We drawing knowLook! that the We can solve this problem by a tree the numbers add up to 1. It rains probability is 0.2 diagram. 0.2 There 0.7 are twoIt does not rain Look!Now Vertically, lets look the at possible events here; rains or It does that itnot rains numbers Tuesday

add up to 1. TheItprobability on rain Tuesday was given to us as rains 0.3.0.3 We can workItout that the 0.8 probability ofand itprobability? not It does Well, What raining isLook! the Vertically, notraining raining the hasMutually to be 0.7, because they not are numbers Exclusive addEvents. up to 1. 1 0.2 = 0.8

have to probabilities add up to 1. have to rain So their We now have the add up to 1. 0.7 required Tree Diagram. It does not rain We wanted to know the probability that it rained on Monday and Tuesday. Tuesday Monday 0.3 It rains 0.7 It does not rain 0.3 It rains 0.2 x 0.3 = 0.06 It rains 0.2 0.8 It does

not rain 0.7 It does not rain We work out path the that probability ofon both events by multiplying the individual This So can the is the probability only through it rains the tree Monday whichhappening and gives Tuesday us rain ison 0.06 both days probabilities together

Actually, we can work out the probabilities of all the possible events Tuesday Monday 0.3 It rains 0.7 It does not rain 0.2 x 0.7 = 0.14 0.3 It rains It rains 0.2 0.8 It does not rain 0.7 It does not rain The probability of rain on Monday, but no rain on Tuesday is 0.14 Tuesday Monday 0.3 It rains

0.7 It does not rain 0.3 It rains It rains 0.2 0.8 It does not rain 0.8 x 0.3 = 0.24 0.7 It does not rain The probability that it will not rain on Monday, but will rain on Tuesday is 0.24 Tuesday Monday 0.3 It rains 0.7 It does not rain

0.3 It rains 0.7 0.8 x 0.7 = 0.56 It does not rain It rains 0.2 0.8 It does not rain The probability that it does not rain on both days is 0.56 Lets look at the completed tree diagram Tuesday Monday 0.3 It rains 0.06 0.7 It does not rain 0.14 0.3

It rains 0.24 It rains 0.2 0.8 It does not rain 0.7 0.56 The end probabilities add up to 1. Remember this! It can help you What your do you notice? check answer! It does not rain 1. There are 2 sets of traffic lights on a journey. A car approaches the first set where the probability of stopping is 0.7. The car then gets to the second set where the probability of stopping is 0.4 a) Draw a tree diagram b) Find the probability of: i. Stopping at both sets of lights ii. Only stopping at one set 2. Julie and Pat are going to the cinema. The probability that Julie will arrive late is 0.2. The probability that Pat will arrive late is 0.6. The two events are independent. a) Draw the tree diagram for this data

b) Find the probability that: i. both will arrive late ii. neither will arrive late 2. Julie and Pat are going to the cinema. The probability that Julie will arrive late is 0.2. The probability that Pat will arrive late is 0.6. The two events are independent. a) Draw the tree diagram for this data b) Find the probability that: i. ii. both will arrive late neither will arrive late 3. Salika travels to school by train every day. The probability that her train will be late on any day is 0.3 (a) Draw the probability tree diagram for Monday and Tuesday. (b) Work out the probability that she is not late on either day c) Work out the probability that she is late on at least one of these days Two balls are chosen from a bag WITHOUT REPLACEMENT. The bag contains 5 red and 9 blue balls. a) Draw the tree diagram b) Find the probability of choosing two red balls c) Find the probability of choosing at least one blue ball. 3. Salika travels to school by train every day. The probability that her train will be late on any day is 0.3 (a) Draw the probability tree diagram for Monday and Tuesday.

(b) Work out the probability that she is not late on either day c) Work out the probability that she is late on at least one of these days 4. Jacob has 2 bags of sweets. Bag P contains 3 green sweets and 4 red sweets. Bag Q contains 1 green sweet and 3 yellow sweets. Jacob takes one sweet at random from each bag. a) Draw the tree diagram for each bag b) Work out the probability of picking two green sweets c) Work out the probability of getting at least at least one green sweet 4. Jacob has 2 bags of sweets. Bag P contains 3 green sweets and 4 red sweets. Bag Q contains 1 green sweet and 3 yellow sweets. Jacob takes one sweet at random from each bag. a) Draw the tree diagram for each bag b) Work out the probability of picking two green sweets c) Work out the probability of getting at least at least one green sweet 5. There are 12 girls and 8 boys in a group. Two are chosen at random by producing names out of a hat. The chosen two are then required to give a presentation to the rest of the group. (a) Draw a tree diagram (b) Find the probability of choosing (i) 2 girls (ii) 1 boy and 1 girl in any order